Integrative and Comparative Biology
Integrative and Comparative Biology, volume 54, number 6, pp. 1099–1108
doi:10.1093/icb/icu114
Society for Integrative and Comparative Biology
SYMPOSIUM
Scaling of the Spring in the Leg during Bouncing Gaits of Mammals
David V. Lee,1,* Michael R. Isaacs,* Trevor E. Higgins,† Andrew A. Biewener† and
Craig P. McGowan‡
*School of Life Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154, USA; †Concord Field Station, Harvard
University, Bedford, MA 01730, USA; ‡Department of Biological Sciences, University of Idaho, Moscow, ID 83844, USA
From the symposium ‘‘Terrestrial Locomotion: Where Do We Stand, Where Are We Going?’’ presented at the annual
meeting of the Society for Integrative and Comparative Biology, January 3–7, 2014 at Austin, Texas.
1
E-mail: [email protected]
Synopsis Trotting, bipedal running, and especially hopping have long been considered the principal bouncing gaits of
legged animals. We use the radial-leg spring constant krad to quantify the stiffness of the physical leg during bouncing
gaits. The radial-leg is modeled as an extensible strut between the hip and the ground and krad is determined from the
force and deflection of this strut in each instance of stance. A Hookean spring is modeled in-series with a linear actuator
and the stiffness of this spring krad is determined by minimizing the work of the actuator while reproducing the measured
force-deflection dynamics of an individual leg during trotting or running, and of the paired legs during hopping. Prior
studies have estimated leg stiffness using kleg , a metric that imagines a virtual-leg connected to the center of mass. While
kleg has been applied extensively in human and comparative biomechanics, we show that krad more accurately models the
spring in the leg when actuation is allowed, as is the case in biological and robotic systems. Our allometric analysis of krad
in the kangaroo rat, tammar wallaby, dog, goat, and human during hopping, trotting, or running show that krad scales as
body mass to the two-third power, which is consistent with the predictions of dynamic similarity and with the scaling of
kleg . Hence, two-third scaling of locomotor spring constants among mammals is supported by both the radial-leg and
virtual-leg models, yet the scaling of krad emerges from work-minimization in the radial-leg model instead of being a
defacto result of the ratio of force to length used to compute kleg . Another key distinction between the virtual-leg and
radial-leg is that krad is substantially greater than kleg , as indicated by a 30–37% greater scaling coefficient for krad . We also
show that the legs of goats are on average twice as stiff as those of dogs of the same mass and that goats increase the
stiffness of their legs, in part, by more nearly aligning their distal limb-joints with the ground reaction force vector. This
study is the first allometric analysis of leg spring constants in two decades. By means of an independent model, our
findings reinforce the two-third scaling of spring constants with body mass, while showing that springs in-series with
actuators are stiffer than those predicted by energy-conservative models of the leg.
Introduction
The spring-mass model is a well-accepted construct
used to characterize oscillations of the center of mass
(CoM) during trotting, running, and hopping
(Blickhan 1989; McMahon and Cheng 1990; Farley
et al. 1993; Blickhan and Full 1993). By abstracting
these ‘‘bouncing’’ gaits to a pogo-stick acting between the CoM and the ground, the spring-mass
model imagines a single virtual leg that acts as a
linear spring between the CoM and the substrate.
Vertical oscillations are coupled with forward
progression such that the virtual-leg rotates over
the foot as it compresses and re-extends, hence,
spring-loaded inverted pendulum (SLIP) is used interchangeably with spring-mass model. It is important to recognize that the spring-mass model or SLIP
is a construct of a point-mass model that is energetically conservative and does not include the geometry, built-in springs, actuators, and dampers of actual
physical legs.
The stiffness of the leg in a simple spring-mass
model often has been estimated as the virtual-leg
spring constant kleg (McMahon and Cheng 1990)
and this metric has been compared quantitatively
to spring constants from SLIP-based simulations of
Advanced Access publication October 9, 2014
ß The Author 2014. Published by Oxford University Press on behalf of the Society for Integrative and Comparative Biology. All rights reserved.
For permissions please email: [email protected].
1100
D. V. Lee et al.
bipedal running (Blum et al. 2009). kleg is the springconstant of an energy-conservative virtual-leg spring
with symmetry about mid-stance. Equation (1) gives
the formula for kleg , the ratio of maximum vertical
force Fmax to the estimated maximal deflection of the
leg, L:
kleg ¼ Fmax =L
ð1Þ
L is estimated from initial leg length L0, maximum vertical deflection of the CoM y, and the
initial angle of the virtual leg with respect to vertical:
L ¼ y þ L0 ð1 cosÞ
ð2Þ
In Equation (3), is in turn estimated from (i)
initial length of the leg L0 in the first instance of
stance, (ii) duration of stance tc, and (iii) average
forward velocity during stance u.
utc
¼ sin1
ð3Þ
2L0
The virtual-leg construct yielding kleg has fueled
key comparative studies of legged locomotion of arthropods and vertebrates spanning four to six orders
of magnitude in body mass (Blickhan and Full 1993;
Farley et al. 1993). kleg has long been recognized to
scale as the two-third power of body mass (Farley
et al. 1993) or, equivalently, spring constant is independent of body mass when normalized by body
mass and leg-length (Blickhan and Full 1993).
These empirical results agree remarkably well with
the predictions of dynamic similarity (Alexander
and Jayes 1983) that maximum force should scale
in direct proportion with body mass and that the
leg should deflect by a constant proportion of leglength, which in turn scales as body mass to the onethird power when animals trot, hop, or run at equal
dimensionless speeds. Hence, two-third scaling of kleg
should be expected from Equation (1):
kleg ¼ Fmax =L / m1 =m1=3 ¼ m2=3
ð4Þ
Empirical scaling relationships for Fmax and L
deviate slightly yet reinforce the predicted two-third
scaling of kleg with body mass (Farley et al. 1993):
m0:67 ¼
m0:97
m0:30
ð5Þ
The two-third scaling of kleg should not be considered an independent result because L is estimated from initial leg-length, L0 (Equations (2)
and (3)). Given the similar exponents of L0 / m0:34
and L / m0:30 , L is only influenced subtly by the
scaling of the other terms in Equations (2) and (3).
Empirical scaling exponents of these parameters are
provided by Farley et al. (1993):
y / m0:36
u / m0:12
/ m0:034
tc / m0:19
ð6Þ
ð7Þ
ð8Þ
ð9Þ
Scaling of vertical deflection of the CoM L was
calculated in Equation (6) from the scaling of maximum vertical force Fmax / m0:97 and that of the
vertical spring constant kvert / m0:61 . The scaling exponent of L will increase with that of according
to Equation (2) such that a greater angle to the vertical yields a greater deflection of the virtual leg. The
scaling exponent of will, in turn, increase with
those of u and tc, but will decrease with the exponent
of L0, according to Equation (3). In sum, the scaling
of kleg as mass to the two-third power is primarily a
consequence of the direct proportionality of force
with mass and the isometric scaling of leg-length as
mass to the one-third power.
The kleg metric has been widely used since its introduction (McMahon and Cheng 1990), in large
part because it was the first method for quantifying
the spring constant of the leg and because it can be
calculated from force-plate data and two additional
measurements: initial length of the leg L0, and initial
forward velocity u. However, the kleg construct has
several shortcomings: (i) kleg is determined from a
single measurement of force and an estimated deflection of the leg, instead of being determined from the
dynamics of the physical legs throughout the stance
and (ii) the assumption of symmetry is typically violated as asymmetrical ground reaction force (GRF;
Table 1) is the norm during bouncing gaits (e.g., Lee
et al. 2004; Cavagna 2006; McGowan et al. 2012;
Andrada et al. 2013; Maykranz and Seyfarth 2014).
In contrast, we consider the spring constant of the
physical leg during the non-conservative, asymmetrical dynamics that are measured experimentally as
time-varying force and deflection throughout each instance of stance. The physical leg is modeled as a
radial strut extending from the hip to the ground,
and the radial-leg’s spring constant krad (Table 1) is
determined from the radial shortening and lengthening of the leg and the force in line with the radial axis
of the leg. krad is determined iteratively based upon a
serial actuator-spring model that minimizes the total
mechanical work of the actuator while reproducing
the experimentally measured force-deflection dynamics of the leg (Lee et al. 2008). Hence, this metric is
distinct in (i) allowing for non-conservative mechanics, (ii) analyzing every instance of stance, (iii) being
1101
Scaling of the spring in the leg during bouncing gaits
Table 1 Notation
Symbol
Unit
Definition
h
m
Height of the hip during standing
g
ms2
1
Acceleration of gravity on Earth
v
pffiffiffiffiffi
Fr
ms
GRF
N
Ground reaction force—the resultant force exerted by the ground on the foot
(trotters and runners) or feet (hoppers) during stance
Fn
N
Normal component of GRF
Fs
N
Shear component of GRF
Mean forward velocity of the trunk during stance
Square-root of the Froude number (Equation (11))
Frad
N
Component of GRF in-line with the radial leg
F rad
N
Mean of Frad during stance
jn
N-s
Time-integral of Fn during stance
js
N-s
Time-integral of Fs during stance
impulse
deg
Mean angle of the GRF impulse during stance (Equation (10))
Lrad
m
Length of the radial leg in each instance
L rad
m
Mean of Lrad during stance
1
Vrad
ms
Vact
ms1
Time-derivative of the serial actuator’s length
jW jrad
J
Total work magnitude of the radial leg (Equation (12))
jW jact
J
Total work magnitude of the actuator in the radial leg (Equation (13))
ARrad
krad
Time-derivative of Lrad
Actuation ratio of the radial leg (Equation (14))
Nm1
Radial-leg spring constant yielding minimum jW jact and ARrad
determined by minimization of actuator work, and
(iv) analyzing the physical leg instead of a virtual
leg originating at the CoM.
Methods
Subjects and experimental procedures: kangaroo rats
and wallabies
Three tammar wallabies (Macropus eugenii) and three
desert kangaroo rats (Dipodomys deserti) hopped
bipedally along level runways while GRF and CoM
trajectory were recorded. Table 2 provides body mass
and leg-length for both species. All testing protocols
were approved by the Harvard University (wallabies)
or University of Idaho (kangaroo rats) Institutional
Animal Care and Use Committees (IACUC). GRF of
a pair of hind limbs was recorded with force plates at
1000 Hz (for wallabies: Kistler type 9286AA, Kistler
Instrument Corp., Novi, MI, USA; for kangaroo rats:
AMTI HE6X6, Advanced Mechanical Technologies
Inc., Watertown, MA, USA). Trials were filmed in
the sagittal plane with digital high-speed video cameras (125 Hz for the wallabies and 200 Hz for the
kangaroo rats). White paint was used to mark the
joints of the hind limbs, as shown in Fig. 1. Markers
were tracked and digitized. The four trials nearest to
steady-speed hopping were selected from each
subject.
Subjects and experimental procedures: dogs and
goats
Level trotting was recorded from three adult dogs (Canis
lupus familiaris) and three adult goats (Capra hircus) on
an indoor runway (10 m long) with a hard rubber surface. Table 2 provides body mass and leg-length for both
species. A pair of rectangular force-platforms (AMTI
BP400600HF, Advanced Mechanical Technologies
Inc., Watertown, MA, USA) were embedded flush
with the runway and oriented end-to-end with long
axes parallel to the runway. All testing protocols were
approved by Harvard University’s IACUC committee.
GRFs of the forelimbs and hind limbs were sampled at
2400 Hz for 5 s and these data were saved using a
National Instruments DAQCardTM 6036E and
LabViewTM 7.1 (National Instruments, Austin, TX,
USA). A QualisysTM 3-D infrared system (3
ProReflexTM MCUs) and Qualisys Track ManagerTM
1.6 software (Qualisys North America, Inc., Highland
Park, IL, USA) were used for capturing motion.
Retroreflective polystyrene hemispheres were used to
mark the forelimb and hind limb joints, lateral
hooves/paws, iliac crests, and the dorsal spine mid-way
1102
D. V. Lee et al.
Table 2 Body mass and basic parameters SDa
Species
m (kg)
L rad (m)
pffiffiffiffiffi
Fr
impulse (deg)
Kangaroo rat—Dipodomys deserti
0.107 0.014
0.054 0.002
2.45 0.36
0.21 0.69
Tammar wallaby—Macropus eugenii
6.42 0.48
0.284 0.015
2.48 0.74
2.33 4.11
Dog—Canis lupus familiaris
23.5 0.8
0.529 0.035
1.19 0.11
0.68 1.16
Goat—Capra aegagus hircus
24.7 2.3
0.479 0.025
1.14 0.13
0.59 3.08
Human—Homo sapiens
69.1 7.1
0.794 0.027
0.89 0.08
1.89 2.60
a
L rad is the mean hind limb length during stance in bipeds, whereas L rad is an average of mean forelimb and hind limb lengths in quadrupeds. For
quadrupeds, the mean impulse angle impulse is a weighted average of the forelimb and hind limb (60:40) to account for greater impulse of the forelimb.
interpolation of motion-capture data included a
fourth order (zero lag) Butterworth lowpass filter
(15 Hz) and cubic (third order) spline. The four
trials nearest to steady-speed running were selected
from each subject.
Kinetic and kinematic parameters
Fig. 1 Schematic of early, mid-stance, and late-stance phases illustrated by the hind limb of a wallaby. The radial-leg is defined as
a line between the proximal joint (hip or scapulohumeral joint)
and the foot (for quadrupeds) or projecting to the ground
through the MTP (hoppers) or the ankle (humans). The serial
actuator-spring model matches the force and deflection of the
radial-leg throughout stance. The actuator is depicted as a piston
and the in-series spring is depicted as a coil.
between the limb girdles. The 3-D positions of these
markers were tracked at 240 Hz. The four trials nearest
to steady-speed trotting were selected from each subject.
Subjects and experimental procedures: humans
Three adult human subjects (Table 2) ran at their
preferred speed on a track with a pair of recessed
Kistler types 9281B and 9281CA force-platforms
(Kistler Instrument Corp., Novi, MI, USA). GRFs
from individual legs were sampled at 1000 Hz using
Vicon Nexus software. Informed consent was obtained
prior to participation, as approved by the Internal
Review Board at the University of Nevada, Las
Vegas. Kinematic data were acquired using a
12-camera system (Vicon MX T40-S; 200 Hz; Vicon
Inc., Los Angeles, USA), and 35-point spatial model
(Vicon Plug-in Gait Fullbody). Filtering and
The GRF normal to the force platform’s surface, Fn,
and the component of shear force in the direction of
travel, Fs, were used in a planar kinetic analysis (see
Table 1 for parameter definitions). GRF directed
toward the animal and forward in the direction of
travel are considered positive. Normal impulse jn and
shear impulse js were determined by numerical integration of Fn and Fs over the contact period tc. The
resultant angle impulse of jn and js, defined with respect to normal, is given by
ð10Þ
impulse ¼ tan1 js =jn :
Positive angles indicate propulsion and negative
ones indicate braking.
Kinematic analysis was restricted to the sagittal
plane. Mean velocity in the direction of travel v
was determined from the forward displacement of
the trunk marker during the contact period and normalized as the square root of the Froude number,
pffiffiffiffiffi
pffiffiffiffiffi
Fr ¼ v= gh;
ð11Þ
where g is the acceleration of gravity and h is the
height of the hip when the animal is standing.
The length of the radial-leg Lrad is determined as
the distance from the proximal joint (i.e., hip or
scapulohumeral) and the hoof or second phalanx
of the lateral digit for goats and dogs, respectively,
in each instance of stance. To account for the long
feet of bipeds, the radial-leg was projected from the
hip joint through the metatarsophalangeal (MTP)
joint of kangaroo rats and wallabies or through the
ankle of humans to the ground. The component of
force in-line with the radial leg Frad is determined by
1103
Scaling of the spring in the leg during bouncing gaits
resolving the GRF vector into the radial axis of the
leg in each instance of stance.
Serial actuator-spring model
In this report, the dynamics of the radial-leg(s) are
determined in five different mammalian species
during trotting, hopping, or running. A Hookean
spring in-series with an actuator represents the
action of all of the muscle-tendon systems about
multiple joints (Fig. 1). The serial actuator-spring
model for the radial-leg analyzes force Frad and deflection (i.e., changes in Lrad ) in every instance of
stance. This model is used to determine the spring
constant of the radial-leg and the fraction of work
required from the in-series actuator of the radial-leg.
The total (positive plus absolute value of negative) of
work done by the radial-leg jW jrad and the total
work done by its in-series actuator jW jact are determined from Frad and the velocities of the radial-leg
Vrad and its actuator Vact , respectively:
Z
ð12Þ
jW jrad ¼ jFrad Vrad jdt
and
Z
jW jact ¼
jFrad Vact jdt:
ð13Þ
jW jact is expressed as a fraction of jW jrad to yield a
dimensionless actuation ratio ARrad —a quantity between zero (completely Hookean) and unity (completely actuated). The spring constant determined by
the model is the one which minimizes jW jact and
consequently actuation ratio:
ARrad ¼ jW jact =jW jrad :
ð14Þ
The radial-leg spring constant krad was determined
iteratively by minimizing ARrad . Using the leastsquares regression of Frad on Lrad as an initial estimate of krad , a range of 105 spring constants was
tried at intervals of 1 Nm1 (Lee et al. 2008).
Statistical analysis
In order to determine the relationships of krad , F rad ,
and jW jrad with body mass, these four parameters
were first log-transformed,
then tested in a leastpffiffiffiffiffi
squares model. Fr and impulse were initially included as independent variables and removed if
they were found to be non-significant (P40.05).
The discrete variable Species was included as an independent variable to test whether it replaced the
significance of body mass. If body mass became
non-significant (P40.05), it was removed from the
model and the allometric analysis was discarded in
favor of interspecific comparisons using Tukey’s
Honest
Significant
Difference
(HSD)
test.
Comparisons between dogs and goats of similar
size were also carried out using a least-squares
model and Tukey’s HSD test without body mass as
an independent variable. Parameter values in Tables
2 and 3 are simple means and standard deviations
(SDs) for each species.
Results
Scaling of radial-leg spring constant krad
The radial-leg spring constant krad was determined by
the serial actuator-spring model for kangaroo rats
(0.107 kg) and wallabies (6.42 kg); two mammalian
trotters: dogs (23.5 kg) and goats (24.7 kg); and
humans running bipedally (69.1 kg) (Table 2). krad
scales with an exponent of 0.70 when all five species
are included (P50.0001; R2 ¼ 0.98; dashed line in
Fig. 2A). Therepffiffiffiffiffi
was no significant effect of dimensionless speed Fr or impulse-angle impulse on krad .
Goats are outliers in this allometric relationship,
having a krad of 13.8 kNm1 that is nearly twice the
krad of 7.5 kNm1 found in dogs. It is clear from
Fig. 2A that the krad of goats falls substantially
above the allometric curve determined for the five
mammals included in this study. When goats are
excluded and the allometric relationship of krad is
recalculated with the four remaining species, a scaling exponent of 0.68 0.02 is found (P50.0001;
Table 3 Radial-leg parameters SDa
Species
krad (kNm1)
F rad (kN)
ARrad
Kangaroo rat—Dipodomys deserti
0.206 0.039
0.0026 0.0004
0.42 0.15
Tammar wallaby—Macropus eugenii
3.40 0.50
0.161 0.023
0.39 0.11
jW jrad (J)
0.146 0.031
30.0 7.5
Dog—Canis lupus familiaris
7.51 1.33
0.315 0.038
0.32 0.06
40.7 9.6
Goat—Capra aegagus hircus
13.81 3.10
0.349 0.031
0.37 0.09
23.8 5.4
Human—Homo sapiens
17.98 3.98
0.941 0.072
0.51 0.13
222.5 43.9
a
Hind-limb values are shown for bipeds. For quadrupeds, krad , F rad , and jW jrad are sums of forelimb and hind-limb values and ARrad is a
weighted-average according to the fore-hind distribution of jW jrad .
1104
D. V. Lee et al.
the scaling exponent for krad is not significantly different from two-third (P50.05).
The allometric scaling of the virtual-leg spring kleg
(Farley et al. 1993) is shown as a gray line in Fig. 2A
for comparison with krad . Both these leg-spring constants scale with exponents that are not significantly
different from two-third. Nonetheless, the allometric
equation for krad shows a greater scaling coefficient
of 0.928–0.979 versus 0.715 for kleg ; hence, the radialleg’s spring is stiffer than the virtual leg’s spring at
any body mass. In fact, the allometric curve for kleg
falls well outside of the 95% confidence limits of the
allometric curve for krad (Fig. 2A).
Scaling of actuation ratio of the radial-leg ARrad
The serial actuator-spring model selects a spring constant by minimizing the actuator work. Expressed as
ARrad (Equation (14)), the ratio of total work done
by the actuator to total work done by the radial-leg,
including work of both the actuator and spring,
averaged between 0.32 and 0.42 for dogs, goats, wallabies, and kangaroo rats, in ascending order, but
averaged 0.51 for human runners (Table 3). ARrad
was found to depend upon species to the exclusion
of body mass, so Tukey’s HSD was used to compare
among species. Human runners have a significantly
greater ARrad than dogs or goats (P50.05) but other
pair-wise comparisons are non-significant.
Scaling of mean force of the radial-leg F rad
Fig. 2 Allometric relationship of the radial leg’s spring-constant,
krad (A) and mean radial-leg force F rad (B) versus body mass in
five mammals. Each plotted point is the mean of three subjects
with four trotting/hopping/running steps each. Species and gait
are identified by symbols in the key. In (A), the power function
for radial-leg spring constant krad is shown for all five species
(dashed line; R2 ¼ 0.98) and with the goat excluded (solid line
with shaded region showing 95% confidence interval of the exponent; R2 ¼ 0.99). The power function for the virtual-leg spring
constant kleg (gray line; Farley et al. 1993; R2 ¼ 0.96) is provided
for comparison. In B, the power function for mean force of the
radial-leg spring F rad is shown (solid line with shaded region
showing 95% confidence interval of the exponent; R2 ¼ 1.0)
alongside the power function for peak leg force Fmax (gray line;
Farley et al. 1993; R2 ¼ 0.97).
R2 ¼ 0.99; black line in Fig. 2A). The krad for goats
falls well above the 95% confidence interval of this
power curve, whereas the other four mammals are
contained within the 95% confidence intervals
(shaded region of Fig. 2A). When goats are excluded,
The mean force in line with the leg F rad scaled with
an exponent of 0.97 0.02 (P50.0001) whenpacffiffiffiffiffi
counting for the significant effect of
Fr
(P50.0001; R2 ¼ 0.99) (Fig. 2B; Table 3). This multiple regression model accounts for the relatively
slower speeds of dogs, goats, and humans compared
with kangaroo rats and wallabies (Table 2). Although
numerically close to unity, the scaling exponent of
0.97 is significantly different from direct proportionality of F rad to body-mass (P ¼ 0.005). There was no
significant effect of impulse-angle impulse on F rad .
Scaling of total work of the radial-leg jW jrad
Total work of the radial-leg jW jrad scaled with an
exponent of 1.23 0.08 (P50.0001) and a coefficient
of
1.25 when accounting for the significant effect of
pffiffiffiffiffi
Fr (P50.0001; R2 ¼ 0.97) (Table 3). The scaling
exponent of 1.23 is significantly different from the
four-third scaling exponent (P ¼ 0.005) predicted by
dynamic similarity on the basis of direct proportionality of force and one-third scaling of leg deflection
with mass. There was no significant effect of impulse-angle impulse on F rad .
Scaling of the spring in the leg during bouncing gaits
Comparison of goats and dogs
Given the nearly equal size of the dogs and goats
included in this study (Table 2), a separate statistical
analysis was used to make direct comparisons of
the forelimbs and the hind limbs of these two quadrupeds. Hence, the independent variable body
mass was replaced by the discrete variable Species
(i.e., goat or dog) in this least-squares model.
The mean force in line with the radial
pffiffiffiffiffi foreleg was
influenced by dimensionless speed Fr , while the
moment-arm about the metacarpophalangeal
pffiffiffiffiffi
(MCP) joint was influenced by both
Fr and
impulse ofpffiffiffiffiffi
the forelimb. There was no significant
effect of Fr or impulse on any of the radial hindleg parameters, hence, the statistical model for the
1105
pffiffiffiffiffi
radial foreleg included Fr and impulse plus the discrete variable Species as independent variables,
whereas the statistical model for the radial hind-leg
included only the discrete variable Species.
According to the serial actuator-spring model, krad
is significantly greater in goats than in dogs
(P50.0001), indicating that the radial hind-legs of
goats are 73% stiffer, and their radial forelegs are
145% stiffer than pthose
ffiffiffiffiffi of dogs (Fig. 3A,B). There
is no effect of
Fr on krad of the hind limb
(P ¼ 0.26) or forelimb (P ¼ 0.37). There is no significant difference between goats and dogs in mean
force F rad of the radial hind-leg (P ¼ 0.22) or foreleg
(P ¼ 0.051), nor is there a significant difference in
actuation ratio ARrad of the radial hind-leg
Fig. 3 Comparison of hind limbs (A) and forelimbs (B) of trotting dogs (light gray) and goats (dark gray) of similar body mass. Radialleg spring constant krad is reported for the individual limb, whereas Fig. 2(A) shows the sum of the hind limb’s and forelimb’s spring
constants. The mean moment arm Rjoint of GRF about the three distal joints of the hind limb (A) and forelimb (B) are shown. (C) Total
work jW jrad and actuation ratio ARrad are shown for the radial hind-leg and radial foreleg of dogs (light gray) and goats (dark gray).
Error bars are 95% confidence intervals. Goats and dogs are significantly different (P50.0001) in all parameters that are not labeled
with other values.
1106
(P ¼ 0.32) or foreleg (P ¼ 0.15) (Fig. 3C). Nonetheless, dogs show significantly greater jW jrad than goats
in both the radial hind-leg and foreleg (Fig. 3C).
To address the mechanism of greater stiffness krad
in the limbs of goats compared with dogs of similar
size, the mean moment arm Rjoint of the GRF is determined about each of the three joints between the
foot and hip joint or foot and shoulder (scapulohumeral) joint. In the hind limb, Rjoint of goats are
significantly shorter than those of dogs about the
MTP and ankle joints (P50.0001) but not about
the knee (P ¼ 0.057; Fig. 3A). Moment arms about
the ankle and MTP of goats are, respectively, 72%
and 77% those of dogs. In the forelimbs, Rjoint
of goats are significantly shorter about the MCP,
wrist, and elbow joints (P50.0001; Fig. 3B) and
are, respectively, 29, p
43,
ffiffiffiffiffi and 77% those of dogs.
There is no effect of Fr or impulse on Rjoint about
hind limb and forelimb joints, except about pthe
ffiffiffiffiffi
MCP, which decreases significantly with
Fr
(P ¼ 0.030) and increases significantly with impulse
(P ¼ 0.028).
Discussion
Scaling of radial-leg spring constant, krad
The scaling of spring constants as a two-third power
of body mass is consistent with dynamic similarity
(Alexander and Jayes 1983) as discussed by Farley
et al. (1993). Dynamic similarity predicts that force
scales as m1 and, assuming that the change in leglength is directly proportional to the initial length of
the leg, deflection scales as m1/3, hence, a springconstant representing the ratio of force, and deflection would scale as m2/3. This is true of kleg , as it is
largely determined by the ratio of peak leg-force to
the initial leg-length, so it should not be expected
to deviate much from two-third scaling. In contrast,
krad is determined from the dynamic force-deflection
pattern of the radial leg according to the serial
actuator-spring model. The scaling result of krad
is particularly compelling because it emerges from
a physics-based model instead of depending upon
a prescribed force-to-length ratio.
The geometry of the tendons provides a potential
mechanism for two-third scaling of leg-spring constants. For example, the cross-sectional area of the
triceps surae tendon is known to scale as the twothird power of body mass (Pollock and Shadwick
1994) across mammalian clades. Because limbs are
complex multi-segmented systems, this explanation
assumes that the average cross-sectional area of tendons and aponeuroses spanning multiple joints can
be represented by the cross-sectional area of the
D. V. Lee et al.
triceps surae tendon. Because tendons are almost entirely collagen, their stiffness is primarily determined
by the number of collagen fibrils in parallel, which is
in turn proportional to the cross-sectional area of the
tendon. A notable exception to two-third scaling
among mammals is cross-sectional area of tendons
within artiodactyla. A classic study found threefourth scaling of tendons’ cross-sectional area—a
result consistent with elastic similarity among artiodactyls (McMahon 1975). Hence, an allometric study
of artiodactyls, such as deer, antelope, cattle, goats,
and sheep, might show that leg-spring constants increase as a three-fourth power of body mass because
the tendons of artiodactyls scale according to elastic,
rather than geometric, similarity. This is a potential
explanation for the disparate krad of goats compared
with the allometric relationship in the present study.
A phylogenetic study applying the serial actuatorspring model would be required to support the hypothesis that tendons’ cross-sectional area determines
krad for artiodactyls as well as for mammals at large.
The idea that the spring constants of legs may be
determined by the thickness of collagen ‘‘ropes’’
spanning multiple joints is appealing in its simplicity.
Nonetheless, detailed musculoskeletal modeling
and simulation are necessary to substantiate the
scaling of tendons’ cross-sectional area as a mechanistic explanation for spring constants across species
and size.
Despite the similarity of two-third scaling both in
krad and kleg , the scaling coefficient of krad is 30–37%
greater than that of kleg (Fig. 2A). Hence, the radialleg spring constant krad is about one-third again as
stiff as the virtual-leg spring-constant kleg at any
given body mass. A proximate explanation for this
difference is that some of the deflection of the radialleg is achieved by the actuator in the serial actuatorspring model, which represents the muscle in the leg.
Because they are in series, the spring and the actuator are subjected to the same force. The serial actuator-spring model found that the work of the
actuator was minimized with a stiffer spring than
the virtual-leg model would predict. For example, a
spring that is too compliant would deflect excessively, causing a greater compensatory deflection
and consequently more work from the actuator. If
built-in leg-springs (i.e., tendons) are too compliant
and they are arranged in-series with their actuators
(i.e., muscles), there is a risk that the compensatory
deflection needed will exceed the shortening capacity
of the muscle to ‘‘take up the slack.’’ This would
impose a limit on tendons’ compliance because
joints such as the ankle could hyper-flex or even
collapse to a plantigrade posture due to unchecked
1107
Scaling of the spring in the leg during bouncing gaits
lengthening of the triceps surae tendon under load.
Hence, the radial-leg spring constant krad is substantially greater than the virtual-leg spring constant kleg
and this greater stiffness reduces deflection of the
built-in leg springs under a given load.
It might be possible to make tendons more compliant and thereby reduce the radial-leg spring
constant to match the virtual-leg spring constant
during level, steady-speed locomotion but this
would neglect non-steady tasks such as locomotion
on an incline or decline, propulsion or braking,
jumping or landing, and falling. If the range of compensatory shortening of the muscle is a limiting
factor, it is likely that tendons are made stiff
enough to accommodate the forces expected during
non-steady locomotion and locomotor perturbations.
Using a serial actuator-spring model, a previous
study found that spring constants of the radial
hind-leg and the ankle joint were unchanged across
level, uphill, and downhill locomotion of goats (Lee
et al. 2008). This result is expected given that the
triceps surae tendon is in-series with the ankle extensor muscles and there is no musculotendon system
in-parallel that could modulate stiffness of the ankle
appreciably. It should be emphasized that, regardless
of the change in length of the extensor muscles, the
stiffness about the ankle is limited by the stiffness of
the triceps surae tendon because the actuator and
spring are in-series. That is, the triceps surae
tendon is held at its proximal end by a position-actuator while it interacts freely about the ankle
through its insertion on the calcaneous. The stiffness
about the ankle, in turn, limits the radial-leg spring
constant krad .
In summary, the importance of non-steady locomotion, together with constraints on compensatory
shortening of the muscle to ‘‘take up the slack,’’ provides potential rationale for our present finding that
krad is one-third again as stiff as kleg . In addition to
more accurately modeling biological systems, this insight may aid in the selection of appropriate spring
constants for robotic prostheses and legged robots in
which springs are used in series with actuators to
achieve a range of locomotor tasks.
artiodactyls from goats as steep-terrain specialists,
we can anticipate functional differences based on approximately two-fold greater spring constant krad observed in goats compared with dogs of similar size.
Additionally, as mentioned in the previous section,
anatomical evidence suggests the possibility that artiodactyls’ legs have spring-constants that scale as
mass to the three-fourth versus mass to the twothird for mammals in general.
Realizing that jumps and drops during ascent and
descent of steep terrain entail greater leg-forces and
that musculotendon springs are in-series, it is expected that radial-leg springs should be stiffer in
goats. If this were not the case, stretching of the
tendons could exceed the ability of the muscles to
compensate by shortening and the joints could
hyper-flex. The radial-leg spring constant is indeed
145% greater in the forelimb and 73% greater in the
hind limb of goats compared with dogs (Fig. 3A,B).
This disproportionate increase in stiffness of the
forelimb is consistent with greater forces exerted on
the forelimbs during vertical drops, which often are
dramatic during evasion of predators. Goats diverged
from other artiodactyls just 23 million years ago
(Dong et al. 2013) and came to occupy an extreme
alpine niche. Noting the absence of comparative
data, if one assumes that artiodactyls have unusually
stiff legs, this could be imagined as a locomotor preadaptation for steep terrain.
In an effort to understand the greater leg-spring
constants of goats compared with dogs, we considered the moment arms of GRF about the distal limbjoints. This analysis showed that the moment arms
of goats are significantly shorter about all joints but
the knee. For a given leg-force, closer alignment of
the GRF to these distal joints will reduce the
moment about that joint, and thereby increase the
resistance to angular deflection of that joint. The
reduction of moment-arms appears to be a mechanism by which goats achieve greater radial-leg spring
constants, yet this is not mutually exclusive of potential differences in the cross-sectional areas of the
tendons.
Conclusions
Comparison of goats and dogs
While dogs are excellent long-distance runners and
terrain generalists, goats are specialists on rough,
steep terrain. Given the disparity of these terrestrial
niches, differences might be expected in the mechanics of the limbs of these two quadrupeds. While we
do not have a sufficient survey for a phylogenetic
analysis and, therefore, cannot distinguish goats as
We use a serial actuator-spring model to perform an
allometric comparison across five species and show
that radial-leg spring constant krad scales with the
same two-third exponent predicted by dynamic similarity and as modeled by the virtual-leg springconstant kleg for trotting, hopping, and running
mammals. This not only supports an effect of
body-mass that holds across different modeling
1108
approaches and agrees with dynamic similarity, but it
also offers the first evidence from an independent
model for two-third scaling that is derived from
actual leg dynamics. Based on scaling coefficients,
krad is one-third again as stiff as kleg at any given
body-mass, which may reflect a need for stiffer
built-in springs (i.e., tendons) to sustain greater
forces without excessive deflection during nonsteady locomotion. Goats are significant outliers
with radial-leg spring constants nearly two-fold
greater than those of dogs of similar size, suggesting
that artiodactyls have stiffer legs and/or goats are
specialized for rough, steep terrain. Allometric analyses of artiodactyls are needed to explore these possible explanations. Additionally, comparative data
from a size-range of rodents, macropods, and carnivores are needed to confirm two-third scaling of the
spring constant of legs within clades and to detect
any differences between trotting, hopping, and running gaits.
Acknowledgments
We thank Dayne Sullivan, Tudor Comanescu,
Andrew Nordin, and Joshua Bailey for the help
with the collection of data, and Pedro Ramirez for
animal-care support.
Funding
This work was supported in part by Defense
Advanced
Research
Projects
Agency
grant
BIOD_0010_2003 [BAA #01-42]; from the National
Institute
of
General
Medical
Sciences
[P20GM103440]; from the National Institutes of
Health and by BEACON under NSF Cooperative
Agreement [DBI-0939454]; and by IDeA Networks
of Biomedical Research Excellence [NIH grant nos.
P20 RR016454 and P20 GM103408].
D. V. Lee et al.
References
Alexander R, Jayes AS. 1983. A dynamic similarity hypothesis
for the gaits of quadrupedal mammals. J Zool 201:135–52.
Andrada E, Nyakatura JA, Bergmann F, Blickhan R. 2013.
Adjustments of global and local hind limb properties
during terrestrial locomotion of the common quail
(Coturnix coturnix). J Exp Biol 216:3906–16.
Blickhan R. 1989. The spring-mass model for running and
hopping. J Biomech 22:1217–27.
Blickhan R, Full RJ. 1993. Similarity in multilegged locomotion: bouncing like a monopode. J Comp Physiol 173:
509–17.
Blum Y, Lipfert SW, Seyfarth A. 2009. Effective leg stiffness in
running. J Biomech 42:2400–5.
Cavagna GA. 2006. The landing–take-off asymmetry in
human running. J Exp Biol 209:4051–60.
Dong Y, Xie M, Jiang Y, Xiao N, Du X, Zhang W, Wang W.
2013. Sequencing and automated whole-genome optical
mapping of the genome of a domestic goat (Capra
hircus). Nat Biotechnol 31:135–41.
Farley CT, Glasheen J, McMahon TA. 1993. Running springs:
speed and animal size. J Exp Biol 185:71–86.
Lee DV, Stakebake EF, Walter RM, Carrier DR. 2004. Effects
of mass distribution on the mechanics of level trotting in
dogs. J Exp Biol 207:1715–28.
Lee DV, McGuigan MP, Yoo EH, Biewener AA. 2008.
Compliance, actuation, and work characteristics of the
goat foreleg and hindleg during level, uphill, and downhill
running. J App Physiol 104:130–41.
Maykranz D, Seyfarth A. 2014. Compliant ankle function results in landing-take off asymmetry in legged locomotion. J
Theoret Biol 349:44–9.
McGowan CP, Grabowski AM, McDermott WJ, Herr HM,
Kram R. 2012. Leg stiffness of sprinters using runningspecific prostheses. J Royal Soc Interface 9:1975–82.
McMahon TA. 1975. Using body size to understand the structural design of animals: quadrupedal locomotion. J Appl
Physiol 39:619–27.
McMahon TA, Cheng GC. 1990. The mechanics of running: how does stiffness couple with speed? J Biomech
23:65–78.
Pollock CM, Shadwick RE. 1994. Allometry of muscle,
tendon, and elastic energy storage capacity in mammals.
Am J Physiol Regul Integr Comp Physiol 266:R1022–31.